Time evolution of ADM and CMC center of mass in general relativity

Abstract

It is shown by several authors going back to Huisken-Yau that asymptotically Schwarzschildean time-slices possess a unique foliation by stable constant mean curvature (CMC) spheres defining the so-called CMC center of mass. We analyze how the leaves of this foliation evolve in time under the Einstein equations. More precisely, we prove that, asymptotically, their time evolution is a translation induced by the quotient of their linear momentum and mass, as to be expected from the corresponding Newtonian setting. In particular, the definitions of mass and linear momentum defined by Arnowitt-Deser-Misner (ADM) are compatible with the interpretation of the CMC foliation as the center of mass of the time-slice by Huisken-Yau. Furthermore, we prove that the coordinate version of the center of mass by Arnowitt-Deser-Misner and the coordinate version of the CMC center of mass coincide - without additional conditions on the scalar curvature. This is even true in the sense of existence, i.e. if one of the two exists then so does the other.

19 pages, added two remarks (1.1 and 2.6) concerning the more general asymptotically flat setting

References in corpus (2)

• Quasilocal angular momentum and center of mass in general relativity
• Explicit Riemannian manifolds with unexpectedly behaving center of mass

Cited by in corpus (4)

• Conserved quantities in general relativity: from the quasi-local level to spatial infinity
• Existence and uniqueness of constant mean curvature foliations of general asymptotically hyperbolic 3-manifolds
• Explicit Riemannian manifolds with unexpectedly behaving center of mass
• Energy, momentum, and center of mass in general relativity